JEE Advanced 2012 Paper 1 Mathematics Q11

Mathematics

Problem

Let θ,φ∈[0,2π]\theta,\varphi\in[0,2\pi] be such that 2cos⁡θ(1−sin⁡φ)=sin⁡2θ(tan⁡θ2+cot⁡θ2)cos⁡φ−1,2\cos\theta(1-\sin\varphi)=\sin^2\theta\left(\tan\frac\theta2+\cot\frac\theta2\right)\cos\varphi-1, tan⁡(2π−θ)>0\tan(2\pi-\theta)>0 and −1<sin⁡θ<−32-1<\sin\theta<-\dfrac{\sqrt3}{2}. Then φ\varphi cannot satisfy

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